Algebraic Topology · Problem 28

Homotopy Theory — Allen Hatcher

Mathematics — Topology · zp-times-zp-no-free-action-sphere

Problem

Show that the group

\mathbb{Z}_p \times \mathbb{Z}_p

with

p

prime cannot act freely on any sphere

S^n

, by filling in details of the following argument. Such an action would define a covering space

S^n \to M

with

M

a closed manifold. When

n > 1

, build a

K(\mathbb{Z}_p \times \mathbb{Z}_p, 1)

from

M

by attaching a single

(n+1)

-cell and then cells of higher dimension. Deduce that

H^{n+1}(K(\mathbb{Z}_p \times \mathbb{Z}_p, 1); \mathbb{Z}_p)

\mathbb{Z}_p

0

, a contradiction. (The case

n = 1

is more elementary.)